Seminars and Colloquia by Series

Reproducing Pairs and Gabor Systems

Series
Dissertation Defense
Time
Tuesday, July 8, 2025 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Logan HartGeorgia Institute of Technology

We first investigate reproducing pairs in Hilbert spaces, with a focus on the discrete case. Reproducing pairs generalize frames and consist of two sequences $\Psi$ and $\Phi$, along with a bounded invertible operator $S_{\Psi,\Phi}$. The work examines sequences that are overcomplete by one element—that is, they become exact upon removal of a single element. A central result shows that if such a sequence admits a reproducing partner, the resulting exact subsequence must form a Schauder basis. This implies that systems like the Gaussian Gabor system at critical density, which lacks a Schauder basis, cannot have a reproducing partner. The result is further generalized to sequences overcomplete by finitely many elements.

Next, we introduce exponential reproducing pairs, where the sequences are weighted exponentials. The associated operator $S_{g\gamma}$ acts as a multiplication operator, and necessary and sufficient conditions are established for when a pair $(g, \gamma)$ forms an exponential reproducing pair.

Lastly, by extending a 2012 result of Heil and Yoon, we develop a two-dimensional theory for weighted exponential systems. It characterizes when weighted double exponential systems are minimal and complete, and provides necessary and sufficient conditions for exactness of arbitrary weighted systems.

Zoom Link: https://gatech.zoom.us/j/93221716846

Applications of Neural Networks with Locally Converging Inputs (NNLCI) for Classical and Quantum PDE Solvers

Series
Dissertation Defense
Time
Monday, July 7, 2025 - 11:00 for 2 hours
Location
Skiles 006
Speaker
Harris Cobb

Please Note: zoom link: https://gatech.zoom.us/j/99430137245

We develop a unified framework for improving numerical solvers with Neural Networks with Locally Converging Inputs (NNLCI). First, we applied NNLCI to 2D Maxwell’s equations with perfectly matched‐layer boundary conditions for light–PEC (perfect electric conductor) interactions. A network trained on local patches around specific PEC shapes successfully predicted solutions on globally different geometries. Next, we tested NNLCI on various ODEs: it failed for chaotic systems (e.g., double pendulum) but was effective for nonchaotic dynamics, and in simple cases can be interpreted as a well‐defined function of its inputs. Although originally formulated for hyperbolic conservation laws, NNLCI also performed well on parabolic and elliptic problems, as demonstrated in a 1D Poisson–Nernst–Planck ion‐channel model. Building on these results, we applied NNLCI to multi‐asset cash‐or‐nothing options under Black–Scholes. By correcting coarse‐ and fine‐mesh ADI solutions, NNLCI reduced RMSE by factors of 4–12 on test parameters, even when trained on a small fraction of the parameter grid. Careful treatment of far‐field boundary truncation was critical to maintain convergence far from the strike price. Finally, we demonstrate NNLCI’s first application to quantum algorithms by improving variational quantum‐algorithm (VQA) outputs for the 1D Poisson equation under realistic NISQ‐device noise. Although noisy VQA solutions deviate from classical finite‐difference references and do not converge to true solutions, NNLCI effectively maps these noisy outputs toward high‐accuracy references. We hypothesize that NNLCI implicitly composes the map from coarse quantum outputs to a noisy convergence space, then to the true solution. We discuss conditions for NNLCI to approximate a well‐defined inverse of the numerical scheme and contrast this with Monte Carlo methods, which lack deterministic intermediate states. These results establish NNLCI as a versatile, data‐efficient tool for accelerating solvers in classical and quantum settings.

Improving Averages over the Prime Numbers and Goldbach's Conjecture

Series
Dissertation Defense
Time
Thursday, July 3, 2025 - 13:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yaghoub RahimiGeorgia Institute of Technology

Please Note: The Zoom link to the meeting: https://gatech.zoom.us/j/99340322307

In this thesis, we investigate three related problems at the intersection of analytic number theory and discrete harmonic analysis. Our primary goal is to understand discrete averaging operators over arithmetic sets—discrete analogues of classical continuous operators—and analyze their behavior using tools from harmonic analysis and additive combinatorics. The results deepen our understanding of how analytic and combinatorial techniques interact in the study of primes and other arithmetic structures.

The Zoom link to the meeting: https://gatech.zoom.us/j/99340322307

Classification of knots vs. links in contact manifolds

Series
Geometry Topology Seminar
Time
Thursday, July 3, 2025 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Rima ChatterjeeOhio State University

 

A knot in a contact manifold is Legendrian if it is everywhere tangent to the contact planes. The classification problem in Legendrian knot theory has always generated significant interest. The problem gets a lot more complicated when we consider links. In this talk, I'll survey some of the results in this area and then discuss the classification problem for cable links of uniformly thick knot type.  If time permits, I'll also mention the classification of links in the overtwisted setting. Part of this is joint work with John Etnyre, Hyunki Min, and Tom Rodewald. 

Representation theory of orthogonal matroids

Series
Dissertation Defense
Time
Thursday, July 3, 2025 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 202 and online
Speaker
Tong JinGeorgia Tech

After quickly recalling the established theory on the combinatorics of orthogonal matroids, we define and study basic properties of the extended rank function and the modular tuples in orthogonal matroids. We then prove a weak version of the path theorem concerning the connectivity of circuits. 

Next, we consider representations of orthogonal matroids over fields (and more generally, over tracts) by bases. We then give a few applications, purely using this basis approach, to the representation theory of orthogonal matroids. We also give a different way of representing orthogonal matroids by circuit functions, which is proved to be equivalent to the basis approach. This is based on joint work with Matthew Baker and joint work with Donggyu Kim. 

The final part of the thesis focuses on the rescaling classes of representations. We construct the foundation of an orthogonal matroid, which possesses the universal property that the set of rescaling classes of representations is in one-to-one correspondence with the set of morphisms from the foundation to the target field. We also give explicit generators and relations of the foundation and an algorithm for computations. 

Zoom link: https://gatech.zoom.us/my/tongjinmath?pwd=QzRDalp2ditGL2tVNUozWm1RK1UwUT09

Counting cliques in graphs with excluded minors

Series
Dissertation Defense
Time
Tuesday, July 1, 2025 - 10:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Ruilin ShiGeorgia Institute of Technology

This thesis explores Turán-type extremal problems in graphs that exclude certain minors, focusing on the maximum number of $k$-cliques such graphs can contain. The first part of the thesis studies planar graphs, which forbid $K_5$ and $K_{3,3}$ as minors. We determine the maximum number of edges is in a planar graph that contains no cycle of length k, and establish a general upper bound for the number of edges in a planar graph avoiding $C_k$ for any $k\ge 11$.

The second part addresses the maximum number of $k$-cliques in $K_t$-minor-free graphs. We show essentially sharp bounds on the maximum possible number of cliques of order $k$ in a $K_t$-minor-free graph on $n$ vertices. More precisely, we determine a function $C(k, t)$ such that for each $k < t$ with $t - k \gg \log_2 t$, every $K_t$-minor-free graph on $n$ vertices has at most $n \cdot C(k, t)^{1 + o_t(1)}$ cliques of order $k$. We also show that this bound is sharp by constructing a $K_t$-minor-free graph on $n$ vertices with $C(k, t) n$ cliques of order $k$. This result answers a question of Wood and Fox–Wei asymptotically up to an $o_t(1)$ factor in the exponent, except in the extreme case where $k$ is very close to $t$.

 

Local geometry determines global landscape in low-rank factorization for synchronization: theory and statistical bounds

Series
Applied and Computational Mathematics Seminar
Time
Thursday, May 8, 2025 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005 and https://gatech.zoom.us/j/94954654170
Speaker
Shuyang LingNYU Shanghai

The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, $\mathbb{Z}_2$-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer-Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the degree of freedom within the factorization exceeds the condition number of the ``Laplacian" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Finally, we will discuss the statistical sides of group synchronization by quantifying the uncertainty of both MLE and spectral estimators.

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